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Journal of Physics B Atomic Molecular and Optical Physics

The dynamics of Tonks–Girardeau gas excited by a pulse drive

Jia 佳 Li 李, Yajiang Hao

Published Dec 16, 2024• 0 citations
Publisher

Abstract

Abstract In this paper we study the dynamics of Tonks–Girardeau (TG) gases in a harmonic potential driven by a Gaussian pulse, which is in correspondence with the excitation dynamics of electrons in matter driven by an ultrashort laser pulse. The evolving dynamics of the TG gas are obtained with a Bose–Fermi mapping method combined with numerical techniques. We calculate the evolving dynamics of the occupation distribution of single-particle energy levels, the density distribution and the momentum distribution of the system. It is shown that the system arrives at a dynamically stable state at the end of driving. In the high-frequency regime, TG gases return back to the ground state, while in the low-frequency regime the population inversion exhibits that all atoms occupy high levels.

DOI
10.1088/1361-6455/ad9fc2

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Abstract In this paper we study the dynamics of Tonks–Girardeau (TG) gases in a harmonic potential driven by a Gaussian pulse, which is in correspondence with the excitation dynamics of electrons in matter driven by an ultrashort laser pulse. The evolving dynamics of the TG gas are obtained with a Bose–Fermi mapping method combined with numerical techniques. We calculate the evolving dynamics of the occupation distribution of single-particle energy levels, the density distribution and the momentum distribution of the system. It is shown that the system arrives at a dynamically stable state at the end of driving. In the high-frequency regime, TG gases return back to the ground state, while in the low-frequency regime the population inversion exhibits that all atoms occupy high levels.
In this paper we study the dynamics of Tonks-Girardeau (TG) gases in a harmonic potential driven by Gaussian pulse, which is a correspondence of the excitation dynamics of electrons in matters driven by ultrashort laser pulse. The evolving dynamics of TG gas are obtained with Bose-Fermi mapping method combined with the numerical techniques. We calculate the evolving dynamics of occupation distribution of single-particle energy levels, density distribution and momentum distribution of the system. It is shown that the system arrived at a dynamically stable state at the end of driving. At high-frequency regime TG gases return back to ground state while at low-frequency regime the population inversion exhibits and all atoms occupy high levels.
The momentum distribution (MD) dynamics of a Tonks-Girardeau (TG) gas is studied in the context of Bragg reflections of a many-body wave packet. We find strong suppression of a Bragg reflection peak for a large and dense TG wave packet; our observation illustrates the dependence of the MD on the interactions and wave function symmetry. The MD is calculated from the reduced single-particle density matrix (RSPDM). We develop a method for calculating the RSPDM of a TG gas, which is operative for a large number of particles, and does not depend on the external potential and the state of the system. The method is based on a formula expressing the RSPDM via a dynamically evolving single-particle basis.
In one-dimensional (1D) quantum gases, the momentum distribution (MD) of the atoms is a standard experimental observable, routinely measured in various experimental setups. The MD is sensitive to correlations, and it is notoriously hard to compute theoretically for large numbers of atoms NNN, which often prevents direct comparison with experimental data. Here we report significant progress on this problem for the 1D Tonks-Girardeau (TG) gas in the asymptotic limit of large NNN, at zero temperature and driven out of equilibrium by a quench of the confining potential. We find an exact analytical formula for the one-particle density matrix $\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Psi}}}}^{\ifmmode\dagger\else\textdagger\fi{}}(x)\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Psi}}}({x}^{\ensuremath{'}})\ensuremath{\rangle}$ of the out-of-equilibrium TG gas in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, valid on distances $|x\ensuremath{-}{x}^{\ensuremath{'}}|$ much larger than the interparticle distance. By comparing with time-dependent Bose-Fermi mapping numerics, we demonstrate that our analytical formula can be used to compute the out-of-equilibrium MD with great accuracy for a wide range of momenta (except in the tails of the distribution at very large momenta). For a quench from a double-well potential to a single harmonic well, which mimics a ``quantum Newton cradle'' setup, our method predicts the periodic formation of peculiar, multiply peaked, momentum distributions.
We summarize in the present work exact results obtained for Tonks-Girardeau gases on one-dimensional optical lattices both for the ground state and nonequilibrium dynamics. On the theoretical side, impenetrable bosons offer the opportunity to study strongly interacting systems in one-dimensional lattices exactly, by means of the Jordan-Wigner transformation, and hence contribute to the topic of strong correlations at the center of interest in both condensed matter physics and quantum gases. This motivation is further enhanced by recent experimental realizations of such systems with ultracold atoms. After having shown their universal properties in equilibrium, we concentrate on their nonequilibrium dynamics. It will be shown that, starting from a pure Fock state, quasi-long-range correlations develop dynamically and lead to the formation of quasicondensates with a momentum determined by the underlying lattice. We expect this effect to be relevant for atom lasers with full control of the wavelength. Then, we will show that the free evolution of an initially confined Tonks-Girardeau gas leads to a momentum distribution that approaches at long times that of the equivalent fermionic system, giving rise to a bosonic gas with a Fermi edge, and hence a fermionization that can only be obtained out of equilibrium. Remarkably, although the momentum distribution function of the Tonks-Girardeau gas becomes equal to the one of the fermions, no loss in coherence is observed in the system, as reflected by a large occupation of eigenstates of the one-particle density matrix.
We develop a general method for calculating the characteristic function of the work distribution of the Tonks-Girardeau (TG) gas confined to an arbitrary time-dependent trapping potential. We use the determinant representation of the many-body wave function to characterise the nonequilibrium thermodynamics of the TG gas and obtain exact and computationally tractable expressions in terms of Fredholm determinants for the mean work, the work probability distribution function, the nonadiabaticity parameter, and the Loschmidt amplitude. When applied to a harmonically trapped TG gas, our results for the mean work and the nonadiabaticity parameter reduce to those derived previously using an alternative approach. We next propose to use periodic modulation of the trap frequency in order to drive the system to highly non-equilibrium states by taking advantage of the phenomenon of parametric resonance. Under such driving protocol, the nonadiabaticity parameter may reach large values, which indicates a large amount of work being done on the system, as compared to sudden quench protocols considered previously. This scenario is realizable in ultracold atom experiments, aiding fundamental understanding of all thermodynamic properties of the system, in addition to facilitating possible optimization of high-efficiency quantum heat engine or quantum refrigeration cycles.
We investigate the static and dynamic properties of a Tonks–Girardeau gas in a harmonic trap with an eccentric δ-perturbation of variable strength. For this, we first find the analytic eigensolution of the single particle problem and then use this solution to calculate the spatial density and energy profiles of the many-particle gas as a function of the strength and position of the perturbation. We find that the crystal nature of the Tonks state is reflected in both the lowest occupation number and the momentum distribution of the gas. As a novel application of our model, we study the time evolution of the spatial density after the sudden removal of the perturbation. The dynamics exhibits collapses and revivals of the original density distribution, which occur in units of the trap frequency. This is reminiscent of the Talbot effect from classical optics.